Modelling of environmental engineering and health problems

Chris P.Tsokos

University of South Florida

Tampa, FL 33620, USA

Some of the important and urgent problems our global society is facing are in environmental (global warming), health sciences, and economics, among others. In the issues of the Journal we will present some current research papers that briefly address the problems, along with extensive references for the convenience of the reader. These manuscripts illustrate the usefulness of mathematical and engineering sciences to perform statistical analysis and mathematical modelling to answer relevant questions and have a better understanding of the subject areas. Some concrete approaches are reflected here.

In engineering, reliability analysis is very important in the design and evaluation of electrical, mechanical, technological, biomedical, etc. products. From a parametric point-of-view, the subject area has been extensively studied. For instance, Bayesian approach to the subject area has been reaching significant momentum with respect to its usefulness in obtaining better estimates of the unknowns. The Bayesian approach requires: (1) knowledge of the failure probability distribution that characterizes the behavior of the failure times of a given system; (2) being able to identify the prior probability distribution if relevant information (data) is available; and (3) identifying or choosing a robust loss function. Presently, we study Bayesian reliability analysis using the useful Weibull probability distribution as our failure model, with the scale parameter being treated as a random variable. We further assume the probability distribution of the scale parameter to follow the general uniform, exponential, inverted gamma, and Jeffreys as priors. The Higgins-Tsokos loss function is used because of its robustness over other loss functions.

Developments in reliability studies are closely related to survival analysis, a very important area in health sciences, especially in all types of cancer. We study parametric, nonparametric, Kaplan-Meir, and Cox PH (proportional hazards) survival analysis and modelling. We illustrate the analytical developing process of each of the subject models and evaluate them using both Monte Carlo simulation and real breast cancer data. Finally, we rank the subject models with respect to effectiveness and usefulness.

We further extend the subject study by employing generalized additive regression modelling to extend the scope of the classical Cox-PH model. The present approach provides flexible models for studying the effect of the prognostic factors on the hazard function. Spline theory is used to predict the failure rate. The effectiveness and usefulness of the subject study is illustrated using breast cancer clinical trials data.

Environmental problems, such as global warming, hurricanes, stony coral species, volcanic explosions, etc., are very important and urgent difficulties that our society is facing in different regions of the world. Global warming is a function of two main entities, atmospheric temperature and carbon dioxide, CO₂. In our study we develop a mathematical model taking into consideration all the attributable variables that have been identified and their corresponding response of the amount of CO₂ in the atmosphere in the continental United States. The developed model includes interactions, higher order entities, in addition to individual contributions to CO₂ in the atmosphere, which are included in the present study. The proposed model has been validated with respect to its efficiency and how it can be used to estimate CO₂ in the atmosphere given information of the attributable variables.

Hurricanes are very catastrophic events, both in loss of human lives and economic losses. Thus, it is important to understand the mechanics underlying the birth and pathway (track) of a tropical storm that might become a full force hurricane. In our study we concentrate on the modelling of hurricane force winds, that is, maximum sustained winds related to pressure, location and linear velocity. We were successful in modelling the wind speed within a storm as a function of the contributing entities. The effectiveness and usefulness in implementing the proposed model are given.

Coral reef communities are very important ecosystems in the world. They are home to at least 4,000 species, or almost a third of the world's marine fish species. For example, the Great Barrier Reef of Australia boasts 400 species of coral providing habitat for more than 1,500 species of fish, 4,000 different kinds of mollusks, and 400 species of sponge. In our study we perform parametric and nonparametric analysis of the Shannon-Wiener diversity index using actual environmental data. Previous analysis of the subject problem assumed the data followed the Gaussian probability distribution, which is not correct and it leads to misleading decisions of the subject problem. The findings and methodology used are applicable to perform similar studies around the world.

Volcanic eruptions occur at various regions around the globe and the unpredictable results can be very catastrophic. A key element in such volcanic events is the tephra fallout. Understan-ding its behavior from a probabilistic point-of-view can be very useful. Two bivariate probability distributions are used to characterize the volcanic explosivity index as well as the tephra fallout and a comparison of the two skewed Gaussian distributions is given.

Decision tree analysis is not very well known, but recently it has been shown to play a very significant role in the analysis of medical and engineering problems. In addition, decision tree analysis has been extensively used in areas in the financial world, for example, loan approval, portfolio management, health and risk assessment, insurance claims evaluation, among others. In our study we review the theory behind decision tree analysis, including software develop-ments and illustrate its usefulness by applying the subject area to various real world problems.

Being able to derive the time evolution of a probability distribution is a problem that we often encounter. For example, for physical time, it could be a kinetic reaction study and identifying the time with the number of computational steps gives information of the type of algorithms used in quantum impurity solvers. The subject area is briefly presented and a new analytical procedure using canonical decomposition into orthogonal polynomials, along with characteristic functions leads into an exact large-time limit distributions.

Time series analysis is a highly powerful method for developing accurate forecasts, especially for phenomenon with non-stationary stochastic realizations. For engineering, economic and health science data, we use autoregressive process, moving averages process or a combination of the two processes to develop forecasting models of the subject information. In our research we introduce a k-th day moving average approach to develop a forecasting model of economic information. Using actual data from an S&P 500 stock, we develop a k-th day moving average integrated ARIMA-moving average model for short and long term forecasting of the price of the given stock. Using residual analysis will illustrate the effectiveness of the proposed model and compare the forecast with the corresponding classical model. The process for developing the subject forecasting model for financial data is applicable to time series data from any other discipline.

In conclusion, using the broad spectrum of mathematical sciences, we can obtain a better understanding of the difficult and very complex problems that our global society is facing.

I wish to thank the Editor, G.L.Degtyarev, Co-Editor, L.K.Kuzmina, and the Editorial Board for their kind invitation to be a Guest Editor of International Journal "Problems of Nonlinear Analysis in Engineering Systems".

Chris P. Tsokos, Ph.D., Distinguished University Professor (University of South Florida, USF), President of IFNA, Executive Director of USOP, Chief-Editor of International Journal "Probability & Statistics"; scientific interests area: mathematics and statistics, methods and models for problems in environmental, engineering, medical-biological systems.

ctsokos@usf.edu